Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add the diced onion, carrots, and celery. Cook, stirring occasionally, until softened, about 8-10 minutes.

Stir in the minced garlic and tomato paste. Cook for another 1-2 minutes until fragrant, stirring constantly to prevent burning.

Pour in the vegetable broth and canned diced tomatoes (undrained). Add the dried oregano, dried basil, and bay leaf. Bring the mixture to a simmer, then reduce heat to low, cover, and let it gently simmer for 15 minutes to allow the flavors to meld.

Stir in the diced zucchini, green beans, rinsed cannellini beans, and kidney beans. Increase the heat to medium-low and continue to simmer, uncovered, for another 10-15 minutes, or until the vegetables are tender-crisp.

Add the small pasta to the simmering soup. Cook according to package directions, typically 8-10 minutes, or until al dente. If the soup becomes too thick, you can add a little more vegetable broth or water.

Remove the bay leaf. Stir in the fresh spinach and cook just until wilted, about 1-2 minutes. Season the soup with salt and freshly ground black pepper to taste.

Ladle the hot minestrone soup into bowls. Garnish each serving with chopped fresh parsley and a generous sprinkle of freshly grated Parmesan cheese before serving.